Make the blueberry basil simple syrup: In a small pot over medium heat, combine the sugar and water. Add the blueberries and basil. Bring to a boil and cook, stirring occasionally, until the sugar has dissolved and the syrup is clear. Let cool completely, about 30 minutes. Strain before using. The syrup will keep in an airtight container in the refrigerator for up to 1 month.
Make the cocktail: In a cocktail shaker, combine the lime juice, 1 ounce blueberry basil simple syrup, and the vodka. FIll the shaker with ice, then shake vigorously. Strain the cocktail into a chilled martini glass and garnish a basil leaf and blueberries.